Dot product: properties

The dot product follows the usual algebra of brackets:

• a · b = b · a (order does not matter).
• a · (b + c) = a · b + a · c.
• (ka) · b = k(a · b).

The most useful fact: a vector dotted with itself gives its length squared. a · a = |a|², since the angle with itself is 0 and cos 0 = 1.

So, to find the length of a combination like a + b, square it with the dot product:

|a + b|² = (a + b) · (a + b) = |a|² + 2a · b + |b|²

Example: |a| = 3, |b| = 4, and a is perpendicular to b (a · b = 0). Then |a + b|² = 9 + 0 + 16 = 25, so |a + b| = 5, as Pythagoras would say.

Also (a + b) · (a – b) = |a|² – |b|², like (x + y)(x – y) = x² – y².

Expanding longer brackets works the same way, term by term. (2a + b) · (a – b) = 2|a|² – 2a · b + b · a – |b|² = 2|a|² – a · b – |b|².

If a + b + c = 0: square both sides. |a|² + |b|² + |c|² + 2(a · b + b · c + c · a) = 0. This gives a · b + b · c + c · a directly from the three lengths.

Two inequalities always hold: |a · b| ≤ |a||b| (Cauchy–Schwarz) and |a + b| ≤ |a| + |b| (triangle inequality: one side of a triangle is never longer than the other two together). Equality in the second means the vectors point the same way.